Recall that the Newtonian system (REF ) is an expanded system of the linear saddle-point problem (REF ), and it is clear that (REF ) is indefinite. Moreover, because of the stiffness matrix \(A_{k_m}\)  in its coefficient matrix \(F^{k_m}\) , the system (REF ) is also ill-conditioned. Hence, it is not easy to solve the Newtonian system (REF ). As analyzed in [1]}, there are two types of algorithms that can be used to solve (REF ): the segregated
and coupled (also known as “all at once") methods.
